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### Cauchy Principal Value

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Date: 05/23/2000 at 05:03:43
From: Derek Waldron
Subject: Cauchy Principal Value

Hi Dr. Math,

Can you tell me exactly what the Cauchy Principal Value is? I know
it's used when a singularity lies on the contour, but all the
literature I've read doesn't actually DEFINE what the value is.

Thanks,
Derek Waldron
```

```
Date: 05/23/2000 at 10:07:40
From: Doctor Rob
Subject: Re: Cauchy Principle Value

Thanks for writing to Ask Dr. Math, Derek.

The Cauchy Principal Value of a divergent integral is the limit as the
radius goes to zero of the integral of the function outside a circular
neighborhood of the singularity.

The simple examples are Riemann integrals along the real line. For
example, the integral of y = x + 1/x from -1 to 2 diverges because
there is a singularity at x = 0. Then the CPV of the integral is the
limit of the integral outside the interval (-r,r), as r -> 0:

-r                     2
CPV = lim  INTEGRAL  (x+1/x) dx + INTEGRAL (x+1/x) dx
r->0         -1                     r

-r                    2
= lim [x^2 + ln(|x|)]     + [x^2 + ln(|x|)]
r->0               x=-1                  x=r

= lim r^2 + ln(r) - 1 - 0 + 4 + ln(2) - r^2 - ln(r)
r->0

= lim  3 + ln(2)
r->0

= 3 + ln(2)

If we took, instead, the interval (-r,2*r), we would get a different
limit. This is an artifact of the divergence of the integral.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus

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